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Title: Determination of the Boltzmann constant by dielectric-constant gas thermometry

Author(s): Fellmuth, Bernd; Fischer, Joachim; Gaiser, Christof; Jusko, Otto; Priruenrom, Tasanee;

Sabuga, Wladimir; Zandt, Thorsten

Journal: Metrologia

Year: 2011, Volume: 48, Issue: 5

DOI: 10.1088/0026-1394/48/5/020

Funding programme: iMERA-Plus: Call 2007 SI and Fundamental Metrology

Project title: T1.J1.4: Boltzmann constant: Determination of the Boltzmann constant for the

redefinition of the kelvin

Copyright note: This is an author-created, un-copyedited version of an article accepted for

publication in Metrologia. IOP Publishing Ltd is not responsible for any errors or omissions in this

version of the manuscript or any version derived from it. The definitive publisher-authenticated

version is available online at doi:10.1088/0026-1394/48/5/020

EURAMETSecretariatBundesallee 10038116 Braunschweig, Germany

Phone:Fax:[email protected]

+49 531 592-1960+49 531 592-1969

1

Determination of the Boltzmann constant by

dielectric-constant gas thermometry

Bernd Fellmuth1, Joachim Fischer1, Christof Gaiser1, Otto Jusko1, Tasanee Priruenrom2,

Wladimir Sabuga1, and Thorsten Zandt1

1 Physikalisch-Technische Bundesanstalt (PTB), Abbestrasse 2-12, 10587 Berlin, and Bundesallee 100,38116 Braunschweig, Germany

2 National Institute of Metrology (Thailand) (NIMT), 3/4-5 Moo 3, Klong 5, Klong Luang, Pathumthani 12120,Thailand, guest researcher at PTB

Abstract

Within an international project directed to the new definition of the base unit kelvin, the

Boltzmann constantk has been determined by dielectric-constant gas thermometry (DCGT) at

PTB. In the pressure range from about 1 MPa to 7 MPa, 11 helium isotherms have been

measured at the triple point of water (TPW) by applying a new special experimental setup

consisting of a large-volume thermostat, a vacuum-isolated measuring system, stainless-steel

10 pF cylindrical capacitors, an autotransformer ratio capacitance bridge, a high-purity gas-

handling system including a mass spectrometer, and traceably calibrated special pressure

balances with piston-cylinder assemblies having effective areas of 2 cm2. The value ofk has

been deduced from the linear, ideal-gas term of an appropriate virial expansion fitted to the

combined isotherms. A detailed uncertainty budget has been established by performing

Monte-Carlo simulations. The main uncertainty components result from the measurement of

pressure and capacitance as well as the influence of the effective compressibility of the

measuring capacitor and impurities contained in the helium gas. The combination of the

results obtained at the TPW (kTPW = 1.380654· 10-23 J/K, relative standard uncertainty 9.2

parts per million) with data measured earlier at low temperatures (21 K to 27 K,

kLT = 1.380657· 10-23 J/K, 15.9 parts per million) has yielded a value ofk =

1.380655× 10-23 J/K with uncertainty of 7.9 parts per million.

1. Introduction

In response to the CIPM proposal to give the Boltzmann constantk a fixed value for a

redefinition of the base unit kelvin [1], many projects have been started to measure

independently the value ofk with a target relative uncertainty of order one part in 106 (one

part per million, ppm). Promising methods are dielectric-constant gas thermometry (DCGT)

[2, 3], acoustic gas thermometry [4, 5], noise thermometry [6], and Doppler-broadening

2

thermometry [7, 8]. An overview of these methods is given in [9]. Within the framework of

the iMERA Joint Research Project “Boltzmann constant” (JRP No. T1.J1.4), PTB designed a

new DCGT experimental setup. This decision has been taken in view of the excellent

experimental DCGT results, which were obtained in the low-temperature range [3, 10, 11, 12,

13] and allowed to set up a thermodynamic temperature scale between 2.5 K and 36 K.

The paper presents results of the determination of the Boltzmann constant via the

measurement of DCGT isotherms at the triple point of water (TPW) applying the new

experimental setup. It is organized as follows. In Section 2, the setup consisting of a large-

volume thermostat, a vacuum-isolated measuring system, stainless-steel 10 pF cylindrical

capacitors, an autotransformer ratio capacitance bridge, a high-purity gas-handling system

including a mass spectrometer, and traceably calibrated special pressure balances with piston-

cylinder assemblies having effective areas of 2 cm2 is described. The experimental results

including the uncertainty budget are discussed in detail in Section 3. Finally conclusions are

drawn and an outlook is given.

2. Experimental setup

2.1. DCGT principle

As illustrated schematically in Figure 1, DCGT is based on the idea to replace the density in

the equation of state of a gas by the dielectric constantε [2, 3, 9]. This idea is realized by

measuring the relative change in capacitanceC

(C(p) – C(0)) / C(0) =εr – 1 +εr κeff p ≡ γ (1)

(with εr = ε/ε0) of a gas-filled capacitor, whereκeff is the effective compressibility of the

capacitor,εr and ε0 are the relative dielectric constant of the gas and the electrical constant,

respectively. The capacitanceC(p) is measured at constant temperature with the space

between its electrodes filled with the measuring gas at various pressuresp and with the space

evacuated so thatp = 0 Pa (measurement of isotherms). The compressibility term accounts for

the deformation of the capacitor electrodes due to the gas pressure and is in sufficient

approximation linear top. For a real gas used in DCGT, a combination of the Clausius-

Mossotti expansion and the density virial expansion has to be considered [2, 3] that leads to

an expansion in the dimensionless parameterµ = γ / (γ + 3)

3

p = A1 (µ + A2 µ2 + A3 µ3 …), (2)

with

A1 = (Aε / RT + κeff / 3) -1, (3)

whereR is the molar gas constant andAε = NA α0 / (3ε0) is the molar polarizability with the

Avogadro numberNA and the atomic dipole polarizabilityα0. The non-linear terms with the

coefficientsA2 and A3 contain the second and third density and dielectric virial coefficients

describing the two- and three-particle interaction. A fit of Expansion (2) to an isotherm

measured at a known thermodynamic temperatureT yields, therefore, the ratioAε/R, which

allows to deduce the Boltzmann constant applying the relation

k = (α0 / ε0) / (3 Aε / R). (4)

Figure 1: Schematic sketch of the DCGT setup used at PTB (reference capacitor on the

left, measuring capacitor on the right). Three quantities have to be measured: pressurep,

capacitanceC(p) and temperatureT via the thermometer resistanceR. χ = εr-1 is the

dielectric susceptibility of the gas. The capsule-type platinum resistance thermometers

are inserted into holes in the block or plate made of copper. The inner diameter of the

holes is only 0.1 mm larger than the outer diameter of the capsule. The thermal contact

is improved with the aid of grease, which fills the gap. Three dry pumps are necessary

for pumping the two vessels containing the capacitors and the large vacuum chamber,

which contains the whole measuring system.

4

The application of Equation(4) requires to knowα0 from fundamental principles. But only

for the measuring gas helium, ab initio calculations, providing the necessary uncertainty

below 1 ppm, are available [14, 15]. Extreme demands are caused by the need to measure the

very small electric susceptibilityχ = εr – 1 of helium (it amounts to only 7⋅ 10-5 at the TPW

and 0.1 MPa). The target relative uncertainty of order one part in 106 can be achieved only if

pressures up to 7 MPa are used. But even at these high pressures, it is necessary to apply a

capacitance bridge that allows to measure capacitance changes with an uncertainty relative to

the capacitance value of order a few parts in 109, see Section 2.3. A further challenge of these

measurements is due to the deformation of the capacitor electrodes under the gas pressure

causing a disturbing additional capacitance change. This means, the effective compressibility

κeff has to be determined with the necessary small relative uncertainty of a few parts in 104.

The investigation of systematic error sources requires comparisons ofκeff values of quite

different special 10 pF capacitors. For this reason, both cylindrical capacitors and very large

multi-ring toroidal cross capacitors described in [16] have been developed.

2.2. Thermal conditions within a special thermostat

The measurement of the Boltzmann constant by DCGT requires to determine the temperature

of the gas and thus of the capacitor electrodes traceably to the definition of the base unit

kelvin, i.e. to the temperature of the TPW, with an uncertainty of order one tenth of a

millikelvin. To realise a thermal environment of sufficient quality, a three-level arrangement

has been realised in the new DCGT experimental setup. Each capacitor is surrounded by a

rigid, metallic pressure vessel, which is thermally anchored to the thick central copper plate of

the measuring system. The temperature of the plate is measured with the aid of three capsule-

type standard platinum resistance thermometers, which have been calibrated at the triple

points of mercury and water as well as at the melting point of gallium. Besides the central

plate at the bottom, the system consists of a top plate, four thick rods connecting the two

plates and an isothermal shield, all made from copper. In turn, the measuring system is placed

within a vacuum chamber for thermal isolation. Finally, the chamber is inserted in a huge

liquid-bath thermostat. The thermal conditions within the experimental setup have been

investigated in detail as described in three accompanying papers [16, 17, 18].

The liquid-bath thermostat has an overall volume of the liquid of about 800 l and a central

working volume, in which the vacuum chamber is located, with a diameter of 500 mm and a

5

height of 650 mm. The temperature stability and the temperature field at the boundary of the

working volume without and with the chamber have been investigated carefully under

different experimental conditions. It could be verified that, under optimum conditions, both

the instability and the inhomogeneity of the temperature in the working volume are well

below 1 mK as necessary [17, 18].

Dedicated experiments described in [16, 18] have shown that the temperature of the central

copper plate can be controlled well within one tenth of a millikelvin under steady-state

conditions over long time periods of a few days. Furthermore, the uncertainty component due

to static temperature-measurement errors can be decreased to the same sufficient level.

Special problems are caused by two unavoidable features of the new DCGT experimental

setup. First, during the experiments, the temperature of the capacitor electrodes inside the

pressure vessels cannot be measured directly. It is not possible to place thermometers inside

the vessels because of the requirement to guarantee a purity of the helium gas of 99.99999%.

Second, due to the huge dimensions of the capacitors and the surrounding pressure vessels,

the mass of the measuring system and thus its heat capacity is very large. This makes the

thermal recovery of the system very slow, which is of importance for the measurements of

isotherms. During such measurements, the flow of the measuring gas for changing the

pressure inside the pressure vessel causes warming or cooling and thus temporary temperature

changes. Both features require to investigate the thermal recovery of the system in order to

reduce dynamic temperature-measurement errors. Dedicated experiments have been

performed by simulating the gas-flow-induced temperature changes via the application of heat

pulses, as also discussed in [16, 18]. The experimental results have been compared with

theoretical calculations based both on simple rough models and finite-element methods. The

obtained time constants of the thermal recovery have an order of magnitude of one hour. The

measurement of an isotherm lasts, therefore, at least several days. Most important is the fact

that the investigations yielded an agreement between experiment and theory, which is

sufficient for deducing the temperature of the capacitor electrodes inside the pressure vessels

with an uncertainty of order one tenth of a millikelvin. Under the real experimental

conditions, this theoretical description of the recovery is of course accompanied by a careful

observation of the drift of the capacitance values with time.

6

2.3. Capacitance measurement

Considering the extreme demands concerning the measurement of capacitance changes, a

high-resolution and high-precision autotransformer ratio capacitance bridge has been built and

tested. Its main component is a home-made high-precision 1:1 inductive voltage divider used

in an autotransformer configuration. For balancing the bridge, adjustable in-phase and

quadrature currents can be injected. A detailed uncertainty budget for measuring small

capacitance changes is presented in [19]. Considering correlations between main terms in the

mathematical model, it is shown that it is possible to measure capacitance changes of at most

a few 0.1% with a relative standard uncertainty below one part per million, i.e. with an

uncertainty relative to the capacitance value of order one part per billion. The performed

consideration of correlations requires that the measuring circuit is fully symmetric. For this

reason, the reference capacitor is also located in the measuring system within the vacuum

chamber.

For checking the reliability of the uncertainty estimates experimentally, the results obtained

with the newly developed bridge have been compared with those obtained with the preceding

bridge applied in the past for DCGT measurements in the low-temperature range, see [12] and

the references cited therein. The preceding bridge, containing a variable home-made high-

precision inductive voltage divider with nine decades, is described shortly in [3]. Within the

upper ratio-error limits of the divider of ten parts per billion, no discrepancies have been

found.

2.4. Gas-handling system and purity analysis

The whole gas-handling system was designed as an ultra-high-purity system using metal

gaskets (VCR metal gasket, Swagelok*)), ultra-high-purity stainless-steel tubing (EN

standard number 1.4435 / X2CrNiMo18-14-3), and electro polished internal surfaces (mean

roughness indexRa ≤ 0.25µm) in all parts (tubes, valves etc.). For precise and reliable tube

connections, an orbital arc welding machine was applied. The manifold allows to connect

helium, neon and argon gas bottles to the system. Before entering the measuring capacitor, the

gas flows through a gas purifier (MicroTorr SP70-902, SAES Pure Gas). According to the

specification of the manufacturer, the gas purifier reduces the content of the impurities H2O,

O2, CO, CO2, H2 and non-methane hydrocarbons from rare gases to less than one part per

billion.

7

The outgassing of hydrogen within the stainless-steel manifold including the capacitor

electrodes might be an essential problem in view of the long time periods necessary for

measuring one isotherm. But gold layers lower the hydrogen permeability by more than one

order of magnitude [20]. Consequently, all central parts in contact with the measuring gas

have been coated with a gold layer of approximately 2 µm thickness.

For analysing the composition of the gas both before entering and directly after leaving the

measuring capacitor, a mass spectrometer (GAM 400, InProcess Instruments) is integrated in

the gas-handling system. Its mass resolution ranges from 0.5 amu to 2 amu, with a detection

limit smaller than 300 ppb for H2, 100 ppb for N2, 50 ppb for CO2, 20 ppb for CH4, and for

the most other elements smaller than 10 ppb (ppb parts per billion). A typical impurity content

in the measuring gas helium after being more than one week within the measuring system is:

370 ppb of H2, 45 ppb of CH4, 40 ppb of Ne, about 10 ppb of Kr and Xe. The found impurity

content leads to an uncertainty estimate for the component “Impurities” in the budget for the

determination of the Boltzmann constant, see Section 3.4, of about 2.4 ppm and will be

reduced in the future by using getters and cold traps. The estimate considers two main facts.

All impurities have a polarizability, which is larger than that of helium, i.e. without a

correction, the treatment of the influence of impurities is not symmetric as prescribed by the

Guide to the Expression of Uncertainty in Measurement [21]. On the other hand, the results of

the analysis do not allow to apply a reliable correction (uncertainty too large, use of detection

limits). Thus, an overall maximum estimate has been calculated by summing up the possible

influences of all impurities and dividing the result by the square root of three, which

corresponds to the application of a non-symmetric rectangular distribution.

2.5. Pressure measurement

The goal to measure pressures up to 7 MPa with a relative uncertainty of order 1 ppm is a

challenge because it requires to characterise pressure balances with a unprecedented quality

and even to improve the national standard of PTB significantly. For absolute pressure

measurements in helium up to 7 MPa, a system of special pressure balances, as outlined in

[22, 23], was designed, constructed and evaluated [24]. The system includes two pressure-

balance platforms, three piston-cylinder units (PCUs) with effective areas of 20 cm2 and three

2 cm2 PCUs. Each platform is equipped with automated mass-piece handlers and a 150 kg

mass-piece set allowing cross-float and pressure measurements in absolute mode. The 11

main mass pieces, each of 12.5 kg, were manufactured in accordance with the requirements of

8

recommendation OIML R 111-1 to masses of ClassE1, see [25], concerning the mechanical

and magnetic properties of their materials, density and surface condition. The design of the

PCUs and their mounting was optimised to reduce the pressure distortion coefficients and

mounting-induced deformations. The gap width of the PCUs was adjusted to reach a

compromise between fall rate and sensitivity. In order to perform automated cross floats of

pressure balances with the possibility of using different gases and a high level of cross-float

sensitivity, differential pressure cells were applied to indicate pressure equilibrium.

Traceability of the pressure measurements to the SI base units up to 7 MPa was realised in

two steps. First, the zero pressure effective areas of the 20 cm2 PCUs have been determined

from dimensional measurements. Second, the 2 cm2 PCUs have been calibrated against the

20 cm2 PCUs by cross-float comparisons. The calibration of the mass pieces traceable to the

national mass standards and the accurate determination of the local gravity acceleration did

not cause special challenges.

Dimensional measurements on the required level of uncertainty below 1 ppm were possible

only on the sufficiently large 20 cm2 PCUs, whose operation-pressure range is limited to

0.75 MPa. Enhanced dimensional measurement techniques [26, 27, 28] were applied to

measure diameters, straightness and roundness of the PCUs. For the first time, uncertainties of

three-dimensional data of order 8 nm (pistons) and 16 nm (cylinders) have been obtained.

The calculation of the effective areas and the estimation of their uncertainties required several

efforts, see the detailed discussion in [23]:

• Evaluation of the three-dimensional data applying a least-squares method [29],

• Use of a two-dimensional flow model to consider the axial non-symmetry [30],

• Consideration of the gas-flow conditions in the clearance [31],

• Measurement of the elastic constants of the PCU’s materials [32],

• Calculation of the pressure distortion coefficient by FEM [33],

• Estimation of the uncertainty of the dimensional data as introduced in [26].

The calculations yielded a relative uncertainty of the effective areas of the three 20 cm2 PCUs

between 0.5 ppm and 0.7 ppm. The 2 cm2 PCU No. 1342 used for the DCGT measurements at

the TPW has been calibrated by cross floating with the 20 cm2 PCUs. The resulting overall

relative uncertainty of the calibration amounts to 1.5 ppm for the zero-pressure effective area,

9

see the detailed budgets presented in [23]. Unfortunately, the cross-float comparison of PCU

No. 1342 with another 2 cm2 PCU at 7 MPa yielded an unexpected difference in pressure

dependence of the effective areas. This may be caused by an error in the calculation of the

pressure distortion coefficient. Applying a rectangular distribution, it causes an additional

relative uncertainty component of 0.16⋅ (p / MPa) ppm.

A complete uncertainty budget for the measurement of a pressure of 7 MPa in a DCGT

experiment at the TPW is given in Table 1. The budget is of course dominated by the

components connected with the zero-pressure effective area and the pressure-distortion

coefficient.

Table 1: Uncertainty budget for the measurement of a pressure of 7 MPa using piston-

cylinder unit No. 1342 with an effective area of 2 cm2.

Component u(p) / p ⋅ 106

Zero-pressure effective area 1.5

Pressure-distortion coefficient 1.1

Mass measurement (piston, mass pieces) 0.1

Gravity acceleration 0.1

Temperature measurement 0.2

Verticality of the PCU 0.1

Combined standard uncertainty 1.9

3. Experimental results

3.1. Determination of the effective compressibility of the measuring capacitors

The changes of the dimensions of the measuring capacitor due to the gas pressurep are

considered in Equations (1) and (3) via the effective compressibilityκeff. For a cylindrical

capacitor as used up to now, the changes of the radii of the inner and outer cylinder due to the

pressure cancel out in the formula for the capacitance. Therefore in an ideal design, only the

relative change∆l(p)/l(0) of the length of the electrodes is relevant. For this ideal case, the

effective compressibility would be one third of the bulk compressibility, which is the inverse

of the bulk modulus.

10

Resonant ultrasound spectroscopy (RUS) [34, 35] is an excellent method for measuring the

elastic properties of a material that determine the bulk compressibility. Only specimen’s

normal-mode frequencies of free vibration were used along with the shape and mass to

determine its elastic properties. Measurements were performed on 10 parallelepipeds cut from

the same rod (stainless steel EN standard number 1.4122 / X39CrMo17-1), from which the

capacitor electrodes were manufactured. The dimensions of the samples were approximately

17 x 13 x 10 mm3, 13 x 10 x 9 mm3 and 12.5 x 11 x 8 mm3, respectively. Care was taken to

ensure that the mean roughness indexRa of the sample faces is within 0.4 µm, and parallelism

and squareness are within 1 µm. The comparison of the results obtained for samples, which

are different with respect to their dimensions and preparation, allowed to estimate how

representative they are for the material comprising the capacitor electrodes. The

corresponding uncertainty component is not dominant at the level achieved up to now. The

dimensions of the samples and their masses have been measured with uncertainties of 1 µm

and 0.5 µg, respectively. The sample resonances were measured att = 0° C using the system

RUSpec from Magnaflux Quasar. The excitation frequency was scanned approximately from

80 to 460 kHz, measuring the first 60 resonances in each case.

The determination of the elastic constants from the measured resonance frequencies is an

inverse problem, the uncertainty of which is primarily limited by that of the dimensions of the

sample. It starts with calculating theoretically an approximate resonance spectrum (direct

problem), which then is compared with the measured spectrum. In an iteration process, the

input parameters have to be adjusted during each iteration step to minimize the error function,

which is defined as the square sum of the differences between the calculated and measured

frequencies.

In order to estimate the uncertainty reliably, two different methods were used to solve the

inverse problem. Firstly, the direct problem was solved with the aid of a computer code,

which has been developed by Migliori et al. [35], and for minimizing the error function, the

generalized reduced gradient method GRG2 [36] came into force. Secondly, the direct

problem was treated applying a finite-element-method (FEM) eigenfrequencies analysis using

the commercial COMSOL Multiphysics simulation software (version 4.1). For the

minimization of the error function, the Nelder–Mead algorithm (downhill simplex) [37] was

implemented in the FEM software. For both methods, the influence of variations in the input

parameters on the output elastic constants has been analysed. Applying both methods to at

11

least six measurements for each sample revealed a relative standard deviation for the bulk

modulus of about 0.02%. The mean values for the bulk modulus, Poisson’s ratio and Young’s

modulus amount to 168.27(19) GPa, 0.2794(3) and 222.53(10) GPa, respectively. This result

leads to a material-determined value of the effective compressibility of

κeff,mat = -1.9809·10-12 Pa-1 at 0°C with a relative standard uncertainty of 0.14%.

Facing the fact that the capacitor within the pressure vessel is a relatively complicated

geometrical object including electrically isolating pieces and stabilising screws, its effective

compressibilityκeff surely deviates from the ideal value. (The gold plating of the capacitor

electrodes has no significant influence onκeff.) Thus, κeff has to be estimated reliably by

performing a FEM simulation. For this simulation, a three-dimensional capacitor model has

been built applying the COMSOL Multiphysics structural mechanics module. For the

Young’s modulus and Poisson’s ratio, the RUS results were used. The density has been

obtained from dimensional and mass measurements. Hydrostatic pressures up to 7 MPa were

considered to act on the inner surface of the closed pressure vessel and all parts of the

cylindrical capacitor. Finally, to prevent rigid body motions of the overall model, two

additional displacement constraints (one for the horizontal directions and one for the vertical

direction) were applied. By changing the constraints, it was later confirmed that they do not

induce reaction forces. The simulations have been carried out after the mesh was completed

(80768 tetrahedral, 19351 triangular and 2990 edge elements). The FEM simulation yielded

the displacement of the electrodes under pressure, i.e.∆l(p)/l(0). It has been found that due to

the special design of the capacitor and its supporting plate,∆l(p)/l(0) is larger for the outer

electrode than for the inner one. Therefore, to estimate the influence of the difference of the

electrode lengths on the capacitance value, additional FEM simulations had to be performed

with the electrostatic module of COMSOL Multiphysics. The estimation showed that the

capacitance change is dominated by∆l(p)/l(0) of the inner electrode, but around 28.6% of the

length difference between inner and outer electrode has to be taken into account. This leads to

an effective length change of∆leff(p) = ∆linner(p) + 0.286 (louter(p)-linner(p)). The combination

of the two FEM simulations leads to a relative correction of the effective compressibility of

1.69% with an uncertainty of 0.12%, which corresponds to a relative uncertainty of the

Boltzmann constant of about 4 ppm. For future improvement, the influence of a more detailed

model of the capacitor, the number of elements and the use of other element types on the

FEM results will be investigated. Finally, it has to be considered that the compressibility

measured by RUS is the adiabatic one, whereas for the DCGT the isothermal compressibility

12

is needed. The conversion between both compressibility values based on fundamental

thermodynamic relations [38] and thermodynamic properties of the stainless steel used, given

by the manufacturer (Dörrenberg Edelstahl), amounts to 1.4% with an uncertainty of 0.07%.

The resulting effective compressibility amounts toκeff = -2.0369·10-12 Pa-1 with a relative

uncertainty of 0.20%. This result has been corroborated by measurements with neon as

measuring gas employing the polarizability value published in [13] (for details see section

3.3). For further improvement of the determination of the bulk compressibility, the influence

of the surface quality on the RUS signal will be investigated in more detail. Also the

thermodynamic properties of the steel used will be determined individually to reduce the

uncertainty of the adiabatic-isotherm correction.

3.2. DCGT isotherms

A total of eleven isotherms have been measured with helium (nominal purity 99.99999%,

supplier Linde AG) and four with neon (99.9993%, Linde AG) using cylindrical capacitors as

described in [18], but without a cylindrical ground-shield spacer at the top. The isotherms

consist of up to fourteen triplets of temperatureT, pressurep andµ (result of the capacitance

measurement, i.e. in fact measure of the susceptibility, see Section 2.1). The overall number

of triplets amounts to 121 for helium and 56 for neon. The pressure values range from

1.125 MPa to 6.75 MPa with a spacing of 0.375 MPa below 5 MPa and 0.75 MPa above. In

most cases, the pressure was increased during the measurement of an isotherm, but for both

gases, two isotherms were taken decreasing the pressure to check the hysteresis of the

capacitor. During all measurements, the deviation of the temperature from 273.16 K was

smaller than 200 mK. The ratio of the two evacuated capacitors (“zero measurement”) has

been determined regularly over the period of the experiments of four months. This allowed to

observe a long-term drift, which amounts to 8 ppm over the whole period and can be

described by a quadratic function. By the aid of this function it is possible to apply for each

isotherm an individual drift correction with an uncertainty of a few ppb.

The sequence of an isotherm measurement with increasing pressure was as follows. After the

“zero measurement”, gas was poured slowly into the measuring capacitor until the lowest

pressure was reached. For measuring the pressure continuously, a quartz-crystal-resonator

sensor is connected to the pressure line. The gas movement caused an increase of the

temperature by a few millikelvin. Due to the slow thermal recovery of the system, see

Section 2.2, it was necessary to wait six to eight hours to reach thermal equilibrium. Then, the

13

pressure balance was connected to the pressure line. The final measurement of a triplet lasted

about ten minutes, whereas the readings of the thermometer, quartz sensor and the null

indicator of the capacitance bridge were averaged over one minute and the mean values stored

on a computer. Figure 2 shows the typical relative scattering of the three quantities

temperature, pressure and susceptibility around the respective mean value over the final-

measurement period. The scatter of the susceptibility must be the largest one because it

detects also the scatter of all other quantities.

0 2 4 6 8 10 12-2

-1

0

1

2

3

4

∆x/x

/ppm

time / min

T - Measurement (STD 0.06 ppm)p - Measurement (STD 0.6 ppm)χ - Measurement (STD 1.4 ppm)

Figure 2: Typical relative scattering∆x / x of the three measurands temperatureT (red

circles), pressurep (blue squares), and susceptibilityχ (black triangles) around their

respective mean values over the final period for measuring one triplet ofT, p andχ on a

DCGT isotherm. The standard deviations (STDs) are given in the legend.

Since the amount of gas poured into the capacitor is smaller for the following pressure steps,

the resulting temperature increase has an order of magnitude of one millikelvin. This caused

recovery periods of about three hours, i.e. the measurement of an isotherm with fourteen

triplets lasted one working week.

To investigate the hysteresis of the measuring capacitor, isotherms have been measured with

increasing and decreasing pressure, respectively. For each of the 10 pressure values, the mean

of the two resultingµ values, which differ both due to hysteresis and non-repeatability, has

14

been calculated. Figure 3 illustrates the results obtained. For the isotherm measured with

increasing pressure, it shows the deviation of theµ value from the meanµ value in

dependence on pressure. The long-term drift of the “zero-measurement” has been corrected

for. The maximum hysteresis amounts to about 20 ppm. Since the hysteresis is mostly

smooth, it can be well described by the fit function Equation (2).

2 3 4 5 6 7-10

0

10

20

∆µ/µ

/ppm

p / MPa

Figure 3: Half relative hysteresis of the DCGT capacitor in the pressure range from

about 3 MPa to 7 MPa:∆µ = µup – (µup + µdown)/2 is the deviation of the valueµupobtained during the measurement of a DCGT isotherm with increasing pressure from

the mean value (µup + µdown)/2 resulting from both isotherm measurements with

increasing and decreasing pressure, respectively.

15

3.3. Evaluation of the DCGT isotherm data

The evaluation of the data started with the correction caused by the long-term drift of the

“zero-measurement” and the transformation to 273.16 K. Then for checking purposes, single-

isotherm fits were made applying Equation (2) with orders from three to five. It has been

found that the fits of third order are most stable in accordance with the magnitude of the virial

coefficients of helium at the TPW. For the therefore preferred order three, Figure 4 shows the

single-isotherm fit residuals of all triplets measured with helium. Below the preferred pressure

range from about 3 MPa to 7 MPa, the scattering is larger due to the small susceptibility of

the gas.

0 1 2 3 4 5 6 7-40

-20

0

20

40Fit residuals of single points

∆p/p

/ppm

p / MPa

Figure 4: Relative single-isotherm fit residuals of all measured helium triplets for the

preferred order three of the fit function Equation (2). Below the preferred pressure range

from about 3 MPa to 7 MPa, the scattering is larger due to the small susceptibility of the

measuring gas helium.

The final fits were made considering all triplets together, either as individual data points or as

meanµ values for the fourteen pressure values. The comparison of the results obtained on

both paths for order three yielded an indication for their robustness in addition to the Monte-

Carlo simulations, see below. As an additional, optional check, the virial coefficients resulting

from the fit parameters of the second and third order have been compared with theoretical

values for the volume virial coefficients given in [39, 40, 41]. The differences between

16

experiment and theory increase smoothly with increasing pressure. These differences have no

influence on the value ofA1, representing the ideal-gas behaviour at low pressures, but will be

investigated more in detail later on with respect to the influence of the dielectric virial

coefficients and the effective compressibility. Finally, the robustness has been investigated by

decreasing the number of the meanµ values included in the fits. This has been done

beginning both at low and at high pressures. The results are shown in Figure 5. It can be seen

that ten data points are sufficient for obtaining stable results for the fitting parameterA1 with a

standard deviation of 3 ppm. For ten data points, the covered pressure ranges are 1.125 MPa

to 4.50 MPa and 2.627 MPa to 6.75 MPa, respectively. It is remarkable that theA1 values

obtained at lower pressures coincide well with the higher pressure ones. This means, the

larger scattering of the data at low pressures is fully statistical.

6 9 12 15-50

0

50

100

150

∆k/k

/ppm

Number of points in the fit

p-range reduced from high pp-range reduced from low p

Figure 5: Dependence of the fit results for the Boltzmann constant on the numbern of

data points if the meanµ values are used as input data for the isotherm fit applying

Equation (2). Starting with the maximum number of 14 points,n has been reduced by

excluding points at low (black circles) and high pressures (red squares). In both cases,

∆k / k is the relative deviation from the result obtained forn = 14.

From the fit to the meanµ values of helium at the TPW in dependence on pressure, a value of

the Boltzmann constant of 1.380654· 10-23 J/K has been deduced applying Equation (4) with

ε0 = 8.8541878· 10-12 F/m and He0α = 2.2815133· 10-41 C2 m2 /J [14, 15] (for a detailed

discussion see [42]), which corresponds toHeεA = 5.1725416· 10-7 m3/mol. Using the molar

17

polarizability of neon NeεA = 9.94727· 10-7 m3/mol published in [13], the data obtained with

neon corroborate the applied value of the effective compressibility of the measuring capacitor,

see Section 3.1, yielding an absolutely larger result of -2.056· 10-12/Pa with an uncertainty of

about 0.9 %, which is mainly limited by the relative standard uncertainty ofNeεA of 11 ppm.

3.4. Uncertainty budget

The complete uncertainty budget for the determination of the Boltzmann constantk by DCGT

at the TPW applying Equation (4) is given in Table 2. It has been established in accordance

with the Guide to the Expression of Uncertainty in Measurement [21]. For the DCGT

measurement, the direct access to the relevant uncertainty estimates is hindered because the

final resultAε/R contained in parameterA1 of the fit function Equation (2), see Equation (3), is

gained by fitting. Therefore, the statistical information for estimating the type A components

has been deduced by Monte-Carlo simulations. This allows also to consider the pressure

dependence of the uncertainty sources. The simulations have been performed with data sets

randomized with the standard deviation of the specific quantity. The combined type A

estimate is corroborated by the comparison of the results obtained applying different fits, see

Section 3.3. The type B components consider uncertainty sources, which influence the value

of k systematically.

The main uncertainty components result from the determination of the dielectric susceptibility

via capacitance changes, the instability of the capacitance of the measuring capacitor (long-

term drift, influence of pressure cycling), calculation of the effective compressibility by FEM

using the elastic constants obtained with RUS, see Section 3.1, and the pressure measurement,

see Section 2.5. A complete budget for the measurement of capacitance changes is given in

[19]. The component estimated for the temperature measurement includes static and dynamic

errors within the measuring system, see Section 2.2. For the measuring gas helium, the head

correction due to the gas column is very small. Its overall relative magnitude for the present

DCGT setup amounts only to 8 ppm. Impurities cause errors since their polarizability differs

from that of helium. Hydrogen having a polarizability, which is larger by a factor of about

four, is especially dangerous. Surface layers on the electrodes of the measuring capacitor play

a minor role because they are present both for theC(p) and theC(0) measurement. The

uncertainty of the polarizability of helium is estimated in [42, 15]. A detailed discussion of

the uncertainty components can be also found in [3, 42, 43]

18

Table 2: Uncertainty budget for the determination of the Boltzmann constantk by dielectric-

constant gas thermometry at the triple point of water.

Component u(k)/k ⋅ 106

Monte-Carlo simulations (type A components)

Susceptibility (scatter of capacitance bridge reading) 3.5

Pressure repeatability 1

Temperature instability 0.5

Capacitance instability 5

Type B estimates

Susceptibility measurement (capacitance change) 1

Determination of effective compressibility (RUS, FEM) 5.8

Temperature (traceability to the TPW) 0.3

Pressure measurement (7 MPa) 1.9

Head correction (pressure of gas column) 0.2

Impurities (measuring gas) 2.4

Surface layers (impurities) 1

Polarizability ab initio calculation (theory) 0.2

Combined standard uncertainty 9.2

4. Conclusions and outlook

A value ofkTPW = 1.380654· 10-23 J/K has been determined for the Boltzmann constant. This

has been done by performing dielectric-constant gas thermometry at the triple point of water

applying a new special experimental setup and helium as the measuring gas. The experimental

setup consists of a large-volume thermostat, a vacuum-isolated measuring system, stainless-

steel 10 pF cylindrical capacitors, an autotransformer ratio capacitance bridge, a high-purity

gas-handling system including a mass spectrometer, and traceably calibrated special pressure

balances with piston-cylinder assemblies having effective areas of 2 cm2. The detailed

analysis of the uncertainty sources including Monte-Carlo simulations yielded an estimate for

the relative standard uncertainty of the obtainedkTPW value of 9.2 ppm.

In an accompanying paper [44], the measurement of the Boltzmann constant in the

temperature range from 21 K to 27 K is described. The low-temperature DCGT experiments

were performed using an apparatus, which has a special cryostat as central part for the

realisation of highly stable thermal conditions. The same apparatus was applied for

19

establishing a thermodynamic temperature scale in the range from 2.5 K to 36 K [12]. The

thermodynamic reference necessary for measuringk was obtained via the realisation of the

International Temperature Scale of 1990, ITS-90 [45]. For estimating the deviation∆T = T90 –

T, the careful investigation of∆T in [46] has been used. To avoid correlations between the

new measurements and former DCGT measurements considered in [46], the deviation∆T in

the range between 21 K and 27 K was analyzed for two different scenarios. The first includes

the former DCGT data and the second one neglects them by setting their uncertainty to

practically infinity. In both cases, the necessary continues (linear) function∆T(T90) for the

deviation in the range between 21 K and 27 K was determined by a fit to the∆T values given

in [46] at selected temperatures. The resulting functions∆T(T90) for the two scenarios agree at

a level of order 0.1 mK, and their mean difference has been used for estimating an additional

small uncertainty component. This component has been added in quadrature to the

uncertainties given in [46]. The uncertainty of the realisation of the triple point of neon as

temperature fixed point (T90 = 24.5561 K) of crucial importance has been estimated on the

basis of an international star intercomparison of sealed triple point cells [47]. Considering the

uncertainties ofT90 – T and the realisation ofT90, a standard uncertainty of the thermodynamic

reference of 9.5 ppm has been estimated. Other main uncertainty components resulted from

the measurement of the dielectric susceptibility via capacitance changes, the instability of the

capacitance of the measuring capacitor, the determination of the effective compressibility

from results obtained at higher temperatures and the pressure measurement. The overall

relative standard uncertainty of the valuekLT = 1.380657· 10-23 J/K obtained at low

temperatures is 15.9 ppm.

The weighted mean of the two valueskLT andkTPW of the Boltzmann constant determined at

low-temperatures and the TPW, respectively, amounts tok = 1.380655· 10-23 J/K and has a

relative standard uncertainty of 7.9 ppm. This value differs from the CODATA value of

1.3806504· 10-23 J/K [48] by 3.2 ppm. In view of the overall combined standard uncertainty of

8.1 ppm, the difference between the two values is not significant. Looking at the state-of-the-

art uncertainty level of the most promising methods of primary thermometry [AGT, RIGT,

NT, DBT], it can be stated that, at present, DCGT is the only method with which an

uncertainty of the determination of the Boltzmann constant below 10 ppm has been obtained

without applying acoustic and / or microwave resonators. Such resonators are used for

acoustic gas thermometry[5] and refractive index gas thermometry [49]. This fact is of

20

special importance because DCGT is independent of the other methods. It has quite different

error sources.

For the first determination of the Boltzmann constant by DCGT at the TPW, the two main

uncertainty components are connected with the properties of the measuring cylindrical

capacitor, namely with its instability and effective compressibility. Progress in decreasing

these components significantly is expected by comparing the parameters of capacitors having

quite different designs, see [16], and by using alternative electrode materials, e.g. tungsten

carbide, the compressibility of which is smaller than that of stainless steel by a factor of three.

First stability tests of cross capacitors gave encouraging results. This is important because the

calculation of their effective compressibility should be more accurate. Activities to decrease

the uncertainty of the pressure measurement to a level of 1 ppm are in progress. First of all,

this concerns the collection of more data on the stability of the dimensions of piston-cylinder

assemblies of the pressure balances and the ratios of their effective areas measured by cross

floating. For reducing the influence of impurities, the outgassing from the walls inside the

pressure vessels has to be investigated in more detail. This requires to analyse better the

content of the most dangerous impurity hydrogen by mass spectrometry. Considering the

experience gained during the first DCGT experiments at the TPW, it seems to be realistic to

decrease the relative uncertainty of the Boltzmann constant to a level of only about 2 ppm

within the next two years.

Acknowledgements

The research within the EURAMET Joint Research Project receives funding from the

European Community's Seventh Framework Programme, iMERAPlus, under Grant

Agreement No. 217257 which is gratefully acknowledged. The authors thank Bettina Thiele-

Krivoi and Norbert Haft for assisting in the development of the experimental setup and

performing the measurements. They are also grateful to Thomas Konczak, Steffen Scheppner

and Helga Ahrendt, who took care of the technical part of the project to design, put into

operation and characterise special pressure balances. Support of this research by the

International Graduate School of Metrology (IGSM) Braunschweig is acknowledged.

21

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